In my humble opinion, καλή ανάρρωση would sound even more inappropriate than περαστικά, if addressing, say, a cancer patient.
Καλή ανάρρωση is something you typically say to someone who has already overcome the severest part of an illness and is entering the stage of recovery. Περαστικά is actually more versatile - at least that's how I see it.
Something else you could say to a person that's facing prolonged and more or less unpredictable hardships (be it disease, loss, divorce or what not) is Καλή δύναμη...